For how many integers $n$ between 1 and 11 (inclusive) is $\frac{n}{12}$ a repeating decimal?
Answer: Recall that a simplified fraction has a terminating decimal representation if and only if the denominator is divisible by no primes other than 2 or 5.

The prime factorization of $12$ is $2^2 \cdot 3$. Therefore, $n/12$ terminates if and only if the numerator has a factor of $3$ in it to cancel out the $3$ in the denominator. Since $3$ integers from $1$ to $11$ are divisible by $3$, there are $11-3=\boxed{8}$ integers $n$ for which the fraction is a repeating decimal.